What are embeddings and what are they used for?

// A toy embedding model: every known word has a 6-number vector over the
// concepts [account, payment, food, travel, problem, delivery].
// A sentence vector is the mean of its word vectors, scaled to length 1.
// Real models learn ~1,000 numbers per text from data; the mechanism is the same.
const WORDS = {
  password: [0.9, 0, 0, 0, 0.2, 0], login: [0.9, 0, 0, 0, 0.1, 0],
  reset: [0.6, 0, 0, 0, 0.3, 0], forgot: [0.5, 0, 0, 0, 0.4, 0],
  credentials: [0.9, 0.1, 0, 0, 0, 0], account: [0.8, 0.2, 0, 0, 0, 0],
  refund: [0, 0.9, 0, 0, 0, 0.2], charged: [0, 0.9, 0, 0, 0.2, 0],
  charge: [0, 0.9, 0, 0, 0.1, 0], duplicate: [0, 0.3, 0, 0, 0.4, 0],
  twice: [0, 0.3, 0, 0, 0.3, 0], money: [0, 0.8, 0, 0, 0, 0],
  pizza: [0, 0, 0.9, 0, 0, 0.2], recipe: [0, 0, 0.9, 0, 0, 0],
  flight: [0, 0, 0, 0.9, 0, 0], booking: [0, 0.2, 0, 0.7, 0, 0],
  package: [0, 0, 0, 0, 0, 0.9], late: [0, 0, 0, 0.1, 0.3, 0.6],
  arrived: [0, 0, 0, 0.2, 0, 0.7], track: [0, 0, 0, 0.1, 0, 0.8],
};
const tokens = (t) => t.toLowerCase().split(/[^a-z]+/).filter(Boolean);
function embed(text, perm = [0, 1, 2, 3, 4, 5]) {
  const v = [0, 0, 0, 0, 0, 0];
  for (const w of tokens(text)) if (WORDS[w]) WORDS[w].forEach((x, i) => (v[perm[i]] += x));
  const n = Math.sqrt(v.reduce((s, x) => s + x * x, 0));
  return v.map((x) => x / n);
}
const cos = (a, b) => a.reduce((s, x, i) => s + x * b[i], 0); // unit vectors: dot = cosine
const overlap = (q, d) => { const s = new Set(tokens(d)); return tokens(q).filter((w) => s.has(w)).length; };
const f3 = (x) => x.toFixed(3);

const docs = ["How to reset your password", "Get a refund for a duplicate charge",
  "Track a late package", "Change your flight booking", "Easy pizza recipe"];
const index = docs.map((d) => ({ text: d, vec: embed(d) }));

console.log("1) semantic search: shared words vs embedding cosine");
for (const q of ["I forgot my login credentials", "I was charged twice, give my money back", "my package never arrived"]) {
  const qv = embed(q);
  const ranked = index.map((d) => ({ d, s: cos(qv, d.vec) })).sort((a, b) => b.s - a.s);
  const top = ranked[0].d.text;
  console.log(`  "${q}"`);
  console.log(`    top hit: "${top}"  cosine ${f3(ranked[0].s)}  runner-up ${f3(ranked[1].s)}  shared words with hit: ${overlap(q, top)}`);
}

console.log("2) deduplication: cosine above 0.95 means the same ticket");
const tickets = ["Forgot password, cannot login", "Need to reset my login password", "Pizza recipe please"];
for (let i = 0; i < tickets.length; i++)
  for (let j = i + 1; j < tickets.length; j++) {
    const s = cos(embed(tickets[i]), embed(tickets[j]));
    console.log(`  ${i + 1} vs ${j + 1}: ${f3(s)} ${s > 0.95 ? "duplicate" : "different"}`);
  }

console.log("3) classification: label of the most similar labelled example");
const labelled = [["Reset my account password", "account"], ["Refund the charge", "billing"], ["Track my package", "shipping"]];
for (const t of ["Login credentials not working", "Money charged twice", "Package is late"]) {
  const best = labelled.map(([x, l]) => ({ l, s: cos(embed(t), embed(x)) })).sort((a, b) => b.s - a.s)[0];
  console.log(`  "${t}" -> ${best.l} (${f3(best.s)})`);
}

console.log("4) mixing models: query from model B, documents from model A");
const modelB = [3, 4, 5, 0, 1, 2]; // same knowledge, different coordinate layout
const qb = embed("I forgot my login credentials", modelB);
const wrong = index.map((d) => ({ d, s: cos(qb, d.vec) })).sort((a, b) => b.s - a.s)[0];
console.log(`  top hit: "${wrong.d.text}" cosine ${f3(wrong.s)} -> vectors from different models are not comparable`);